Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Dear Calendar People

On the May 16 this year in reply to the issue of continued fractions I said:

“I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion
goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.”

I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring
circles will touch without overlapping. See

If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar
is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading
to greater jitter and more difficulty working out a conversion algorithm for the calendar.

Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from
other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion.

I found one measure of structural complexity, which involves Ford circles:

The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia
picture.

Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and
are

… 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, …

These cycles have complexity 1.

The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33-year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years
with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions:

8/33 for solar leap day calendars

2/11 & 3/17 for leap week calendars (but not 5/28)

8/15 & 9/17 for months in pure lunar calendars. These are the yerms in my lunar yerm calendar

3/8 & 4/11 for the leap month years in a lunisolar calendar.

You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19-year Metonic cycle with 7 leap years (touches 3/8) and
the Tabular Islamic calendar (touches 4/11). Another example is the 28-year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions:

Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128)

11/62 for leap week calendars (also 5/28, 8/45 & 14/79)

26/49 for months in for pure lunar calendars (3-yerm cycle)

11/30 for 12-month years in pure lunar calendars (but not 29/79)

7/19 for leap month years in a lunisolar calendar

Complexity 4 includes the 334-year cycle, 353-year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293-year leap week cycle used in the Symmetry454 calendar and also the months of my yerm
calendar. Examples previously mentioned have fractions:

52/293 & 69/389 for leap week calendars (also 41/231)

451/850 for months of my yerm calendar or any other one-era cycle

29/79 for 12-month years in a pure lunar calendars

123/334 & 130/353 for the leap month years of a lunisolar calendar.

Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions:

71/400 & 159/896 for leap week calendars

928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multi-era cycle

239/649, 267/725 & 2519/6840 for the leap month years of lunisolar calendars

In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850)
have complexity 4. This is an example of a mixing addition rule (with yerms mixed) that applies to this measure of complexity.

Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either
of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion.

I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3.

cLc|cL|cLc|cLc|cLc|cL|cLc|cLc|cLc|cL|cLc|

Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3.

cLccLcLc|cLccLccLcLc|cLccLccLcLc|

Both steps of mixing can be shown as

cLc|cL|cLc||cLc|cLc|cL|cLc||cLc|cLc|cL|cLc|

Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1.
I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target.

Re: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Dear Calendar People

I list all examples, with mean year 0 to ½ , up to 8 years long, in the order of their mean year.

I’ve started each example cycle, so that the leap years occur as late as possible and so year Y of a C-year cycle with L leap years is I leap year if and only if

Remainder (Y*L) divided by C is less than L.

I’ve also coloured red any leap year preceded by an odd number of common years along with those common years to show the structure.

COMPLEXITY 0

0/1 0.0 ‘c’

COMPLEXITY 1

1/8 0.125 ‘cccccccL’

1/7 0.142857… ‘ccccccL’

1/6 0.166666… ‘cccccL’

1/5 0.2 ‘ccccL’

1/4 0.25 ‘cccL’
Julian Calendar

COMPLEXITY 2

2/7 0.285714… ‘cccLccL’

COMPLEXITY 1

1/3 0.333333… ‘ccL’

COMPLEXITY 2

3/8 0.375 ‘ccLccLcL’
Octaeteris

2/5 0.4 ‘ccLcL’

3/7 0.428571… ‘ccLcLcL’

COMPLEXITY 1

1/2 0.5 ‘cL’

All cycles whose mean year lies between those of any two cycles on consecutive lines (of same complexity) are of greater complexity. There are an infinite number of cycles of complexity 2 with mean year between
a cycle of complexity 1 and an adjacent cycle of complexity 2 and they converge to the cycle of complexity 1.

I reckon the shortest example of complexity 3 has 12 years of which either 5 or 7 are leap

COMPLEXITY 3

5/12 0.416666… ‘ccLcLccLcLcL’

It is the mediant of 2/5 and 3/7 and with leap years placed as late as possible is also the concatenation of these two in the order of their mean year.

This cycle would apply to a 12 month year of 365 days with the 30 & 31 day month spread as smoothly as possible (and as shown here, with the 31-day months as late as possible).

It also corresponds to the white and black notes of a piano keyboard (and as shown here, starting from B).

I reckon the shortest cycles of each level of complexity are

1/2, 2/5, 5/12, 12/29, 29/70, … converging to the fractional part of the square root of two.

The integer number sequence they contain is the Pell numbers in which P(n+1) = 2*P(n) + P(n-1).

From: Palmen,
Karl (STFC,RAL,ISIS) Sent: 14 October 2016 13:04To:[hidden email]Subject: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Dear Calendar People

On the May 16 this year in reply to the issue of continued fractions I said:

“I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion
goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.”

I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring
circles will touch without overlapping. See

If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar
is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading
to greater jitter and more difficulty working out a conversion algorithm for the calendar.

Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from
other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion.

I found one measure of structural complexity, which involves Ford circles:

The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia
picture.

Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and
are

… 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, …

These cycles have complexity 1.

The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33-year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years
with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions:

8/33 for solar leap day calendars

2/11 & 3/17 for leap week calendars (but not 5/28)

8/15 & 9/17 for months in pure lunar calendars. These are the yerms in my lunar yerm calendar

3/8 & 4/11 for the leap month years in a lunisolar calendar.

You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19-year Metonic cycle with 7 leap years (touches 3/8) and
the Tabular Islamic calendar (touches 4/11). Another example is the 28-year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions:

Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128)

11/62 for leap week calendars (also 5/28, 8/45 & 14/79)

26/49 for months in for pure lunar calendars (3-yerm cycle)

11/30 for 12-month years in pure lunar calendars (but not 29/79)

7/19 for leap month years in a lunisolar calendar

Complexity 4 includes the 334-year cycle, 353-year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293-year leap week cycle used in the Symmetry454 calendar and also the months of my yerm
calendar. Examples previously mentioned have fractions:

52/293 & 69/389 for leap week calendars (also 41/231)

451/850 for months of my yerm calendar or any other one-era cycle

29/79 for 12-month years in a pure lunar calendars

123/334 & 130/353 for the leap month years of a lunisolar calendar.

Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions:

71/400 & 159/896 for leap week calendars

928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multi-era cycle

239/649, 267/725 & 2519/6840 for the leap month years of lunisolar calendars

In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850)
have complexity 4. This is an example of a mixing addition rule (with yerms mixed) that applies to this measure of complexity.

Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either
of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion.

I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3.

cLc|cL|cLc|cLc|cLc|cL|cLc|cLc|cLc|cL|cLc|

Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3.

cLccLcLc|cLccLccLcLc|cLccLccLcLc|

Both steps of mixing can be shown as

cLc|cL|cLc||cLc|cLc|cL|cLc||cLc|cLc|cL|cLc|

Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1.
I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target.

I list all examples, with mean year 0 to ½ , up to 8 years long, in the order of their mean year.

I’ve started each example cycle, so that the leap years occur as late as possible and so year Y of a C-year cycle with L leap years is I leap year if and only if

Remainder (Y*L) divided by C is less than L.

I’ve also coloured red any leap year preceded by an odd number of common years along with those common years to show the structure.

COMPLEXITY 0

0/1 0.0 ‘c’

COMPLEXITY 1

1/8 0.125 ‘cccccccL’

1/7 0.142857… ‘ccccccL’

1/6 0.166666… ‘cccccL’

1/5 0.2 ‘ccccL’

1/4 0.25 ‘cccL’
Julian Calendar

COMPLEXITY 2

2/7 0.285714… ‘cccLccL’

COMPLEXITY 1

1/3 0.333333… ‘ccL’

COMPLEXITY 2

3/8 0.375 ‘ccLccLcL’
Octaeteris

2/5 0.4 ‘ccLcL’

3/7 0.428571… ‘ccLcLcL’

COMPLEXITY 1

1/2 0.5 ‘cL’

All cycles whose mean year lies between those of any two cycles on consecutive lines (of same complexity) are of greater complexity. There are an infinite number of cycles of complexity 2 with mean year between
a cycle of complexity 1 and an adjacent cycle of complexity 2 and they converge to the cycle of complexity 1.

I reckon the shortest example of complexity 3 has 12 years of which either 5 or 7 are leap

COMPLEXITY 3

5/12 0.416666… ‘ccLcLccLcLcL’

It is the mediant of 2/5 and 3/7 and with leap years placed as late as possible is also the concatenation of these two in the order of their mean year.

This cycle would apply to a 12 month year of 365 days with the 30 & 31 day month spread as smoothly as possible (and as shown here, with the 31-day months as late as possible).

It also corresponds to the white and black notes of a piano keyboard (and as shown here, starting from B).

I reckon the shortest cycles of each level of complexity are

1/2, 2/5, 5/12, 12/29, 29/70, … converging to the fractional part of the square root of two.

The integer number sequence they contain is the Pell numbers in which P(n+1) = 2*P(n) + P(n-1).

From: Palmen,
Karl (STFC,RAL,ISIS) Sent: 14 October 2016 13:04To: CALNDR-L@...Subject: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Dear Calendar People

On the May 16 this year in reply to the issue of continued fractions I said:

“I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion
goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.”

I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring
circles will touch without overlapping. See

If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar
is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading
to greater jitter and more difficulty working out a conversion algorithm for the calendar.

Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from
other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion.

I found one measure of structural complexity, which involves Ford circles:

The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia
picture.

Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and
are

… 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, …

These cycles have complexity 1.

The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33-year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years
with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions:

8/33 for solar leap day calendars

2/11 & 3/17 for leap week calendars (but not 5/28)

8/15 & 9/17 for months in pure lunar calendars. These are the yerms in my lunar yerm calendar

3/8 & 4/11 for the leap month years in a lunisolar calendar.

You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19-year Metonic cycle with 7 leap years (touches 3/8) and
the Tabular Islamic calendar (touches 4/11). Another example is the 28-year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions:

Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128)

11/62 for leap week calendars (also 5/28, 8/45 & 14/79)

26/49 for months in for pure lunar calendars (3-yerm cycle)

11/30 for 12-month years in pure lunar calendars (but not 29/79)

7/19 for leap month years in a lunisolar calendar

Complexity 4 includes the 334-year cycle, 353-year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293-year leap week cycle used in the Symmetry454 calendar and also the months of my yerm
calendar. Examples previously mentioned have fractions:

52/293 & 69/389 for leap week calendars (also 41/231)

451/850 for months of my yerm calendar or any other one-era cycle

29/79 for 12-month years in a pure lunar calendars

123/334 & 130/353 for the leap month years of a lunisolar calendar.

Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions:

71/400 & 159/896 for leap week calendars

928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multi-era cycle

239/649, 267/725 & 2519/6840 for the leap month years of lunisolar calendars

In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850)
have complexity 4. This is an example of a mixing addition rule (with yerms mixed) that applies to this measure of complexity.

Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either
of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion.

I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3.

cLc|cL|cLc|cLc|cLc|cL|cLc|cLc|cLc|cL|cLc|

Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3.

cLccLcLc|cLccLccLcLc|cLccLccLcLc|

Both steps of mixing can be shown as

cLc|cL|cLc||cLc|cLc|cL|cLc||cLc|cLc|cL|cLc|

Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1.
I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target.

Re: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Dear Walter and Calendar People

The complexity of such a calendar would depend on the leap year rule for the leap months and would apply only if the leap years were spread as smoothly as possible.

I have forgotten what his leap year rule was, but I guess it is 13 leap years in 293 years. The intervals between the leap years alternate between 22 & 23 years,
except for one pair of consecutive 23s. This I reckon to be complexity 3.

Complexity 2 is when all intervals between years of the minority type (leap years) are equal with ONE exception which is ONE year different. Examples are the
33-year cycle of 8 leap years or the 45-year cycle of 2 leap years for a 28-day month calendar.

I think the reason Walter thinks it is complexity 2, may be that he has been looking at the sequence of intervals between the leap years rather than the sequence
common years and leap years. The intervals form two types that are spread as smoothly as possible and so have a complexity and this complexity is 1 less (if the leap years are a minority). This process can be repeated again and again this we have just one
interval and in so doing we measure the complexity.

13/293
ccc…cLc… etc. complexity 3

7/13
23:22:23:22:23:22:23:22:23:22:23:22:23 complexity 2

Taking intervals between the 22s which are the minority type

1/6 3:2:2:2:2:2
complexity 1

Then we have only 1 interval between the 3s (of 6).

The complexity can be measured by means of repeated mixing in which each mix has one of the one type and one or more of another type of equal complexity.

I show this mixing for the 28-day month calendar:

0/1: All years have 13 months, no leap years: Complexity 0.

1/1: All years have 14 months, every year is a leap year: Complexity 0.

1/23: Mix 22 of 0/1 with
1 of 1/1 and we get a 23-year cycle with one leap year: complexity 1.

1/22: Mix 21 of 0/1 with
1 of 1/1 and we get a 22-year cycle with one leap year: complexity 1

2/45: Mix 1 of 1/23 with
1 of 1/22 and we get 45-year cycle with two leap years: complexity 2

3/68: Mix 2 of 1/23 with
1 of 1/22 and we get 68-year cycle with three leap years: complexity 2

13/293: Mix 5 of 2/45 with
1 of 3/68 and we get 293-year cycle with thirteen leap years: complexity 3.

For leap weeks the corresponding cycle is 52/293, which has complexity 4.

0/1: All years have 52 weeks, no leap years: Complexity 0.

1/1: All years have 53 weeks, every year is a leap year: Complexity 0.

1/6: Mix 5 of 0/1 with
1 of 1/1 and we get 6-year cycle with one leap year: complexity 1

1/5: Mix 4 of 0/1 with
1 of 1/1 and we get 5-year cycle with one leap year: complexity 1

3/17: Mix 2 of 1/6 with
1 of 1/5 and we get 17-year cycle with three leap years: complexity 2

2/11: Mix 1 of 1/6 with
1 of 1/5 and we get 11-year cycle with two leap years: complexity 2

11/62: Mix 3 of 3/17 with
1 of 1/11 and we get 62-year cycle with 11 leap years: complexity 3

8/45: Mix 2 of 3/17 with
1 of 1/11 and we get 45-year cycle with 8 leap years: complexity 3

52/293: Mix 4 of 11/62 with
1 of 8/45 and we get 293-year cycle with 52 leap years: complexity 4

The Gregorian leap week cycle 71/400 would require one more step,, because it is 5 of 11/62 and
2 of 8/45 and so has complexity 5 as shown in my original note. Walter’s 400-leap week cycle based on his modified 33-year cycle is an example of this.

Note that these mixes are not determined by continued fraction convergents. Sometimes a convergent may be skipped. Instead they are determined by Ford circles
on which this complexity is defined.

I list all examples, with mean year 0 to ½ , up to 8 years long, in the order of their mean year.

I’ve started each example cycle, so that the leap years occur as late as possible and so year Y of a C-year cycle with L leap years is I leap year if and only if

Remainder (Y*L) divided by C is less than L.

I’ve also coloured red any leap year preceded by an odd number of common years along with those common years to show the structure.

COMPLEXITY 0

0/1 0.0 ‘c’

COMPLEXITY 1

1/8 0.125 ‘cccccccL’

1/7 0.142857… ‘ccccccL’

1/6 0.166666… ‘cccccL’

1/5 0.2 ‘ccccL’

1/4 0.25 ‘cccL’
Julian Calendar

COMPLEXITY 2

2/7 0.285714… ‘cccLccL’

COMPLEXITY 1

1/3 0.333333… ‘ccL’

COMPLEXITY 2

3/8 0.375 ‘ccLccLcL’
Octaeteris

2/5 0.4 ‘ccLcL’

3/7 0.428571… ‘ccLcLcL’

COMPLEXITY 1

1/2 0.5 ‘cL’

All cycles whose mean year lies between those of any two cycles on consecutive lines (of same complexity) are of greater complexity. There are an infinite number of cycles of complexity 2 with mean year between
a cycle of complexity 1 and an adjacent cycle of complexity 2 and they converge to the cycle of complexity 1.

I reckon the shortest example of complexity 3 has 12 years of which either 5 or 7 are leap

COMPLEXITY 3

5/12 0.416666… ‘ccLcLccLcLcL’

It is the mediant of 2/5 and 3/7 and with leap years placed as late as possible is also the concatenation of these two in the order of their mean year.

This cycle would apply to a 12 month year of 365 days with the 30 & 31 day month spread as smoothly as possible (and as shown here, with the 31-day months as late as possible).

It also corresponds to the white and black notes of a piano keyboard (and as shown here, starting from B).

I reckon the shortest cycles of each level of complexity are

1/2, 2/5, 5/12, 12/29, 29/70, … converging to the fractional part of the square root of two.

The integer number sequence they contain is the Pell numbers in which P(n+1) = 2*P(n) + P(n-1).

On the May 16 this year in reply to the issue of continued fractions I said:

“I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months
are spread as smoothly as possible. Chapter 4 the conclusion goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.”

I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal
line each touching the line at position L/C, then neighbouring circles will touch without overlapping. See

If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are
general algorithms to convert such a calendar and such a calendar is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in
which the leap years are not spread as smoothly as possible, leading to greater jitter and more difficulty working out a conversion algorithm for the calendar.

Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern
of the leap years and common years. It is different from other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion.

I found one measure of structural complexity, which involves Ford circles:

The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford
circles are 0/1 & 1/1 and are shown in brown in the Wikipedia picture.

Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding
Ford circles are those that touch those of complexity 0 and are

… 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, …

These cycles have complexity 1.

The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33-year cycle
of 8 leap years (touches ¼) or the Octaeteris of 8 years with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions:

8/33 for solar leap day calendars

2/11 & 3/17 for leap week calendars (but not 5/28)

8/15 & 9/17 for months in pure lunar calendars. These are the yerms in my lunar yerm calendar

3/8 & 4/11 for the leap month years in a lunisolar calendar.

You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the
19-year Metonic cycle with 7 leap years (touches 3/8) and the Tabular Islamic calendar (touches 4/11). Another example is the 28-year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions:

Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128)

11/62 for leap week calendars (also 5/28, 8/45 & 14/79)

26/49 for months in for pure lunar calendars (3-yerm cycle)

11/30 for 12-month years in pure lunar calendars (but not 29/79)

7/19 for leap month years in a lunisolar calendar

Complexity 4 includes the 334-year cycle, 353-year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293-year leap week cycle used
in the Symmetry454 calendar and also the months of my yerm calendar. Examples previously mentioned have fractions:

52/293 & 69/389 for leap week calendars (also 41/231)

451/850 for months of my yerm calendar or any other one-era cycle

29/79 for 12-month years in a pure lunar calendars

123/334 & 130/353 for the leap month years of a lunisolar calendar.

Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions:

71/400 & 159/896 for leap week calendars

928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multi-era cycle

239/649, 267/725 & 2519/6840 for the leap month years of lunisolar calendars

In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm
cycle (17/52) has complexity 2, while the months (451/850) have complexity 4. This is an example of a mixing addition rule (with yerms mixed) that applies to this measure of complexity.

Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any
cycle formed by mixing them is equal to the complexity of either of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion.

I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3.

cLc|cL|cLc|cLc|cLc|cL|cLc|cLc|cLc|cL|cLc|

Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3.

cLccLcLc|cLccLccLcLc|cLccLccLcLc|

Both steps of mixing can be shown as

cLc|cL|cLc||cLc|cLc|cL|cLc||cLc|cLc|cL|cLc|

Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration
raises the complexity of the cycle by no more than 1. I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target.

The Ford circle of a cycle is the Ford circle of the fractional part of its mean year, which will serve as notation of the example cycles below.

For example, the 293-year leap week cycle 52/293 has its Ford circle
touch 11/62, which in turn touches 3/17, which in turn touches 1/6, which in turn
touches 0/1. There are 4 touchings, therefore 4 steps and also this is the minimum and so 52/293 has complexity 4. Note that the Ford circles of a/b & c/d touch if and only if ad & bc differ by 1.

A cycle has complexity 1 if it has only one leap year or one common year.

A cycle has complexity 2 if its number of years is one different from a proper multiple of the number of leap years or the number of common years.

Re: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

In my original note I showed examples grouped according to their complexity. Here I show them according to cycle type. This may better show the relationship between cycles of different complexity. I gain write
each cycle down as the fractional part of its mean year in units of intercalation. The complexity applies only to cycles whose leap years are spread as smoothly as possible.

I use a comma to separate mixer cycles (Ford circles touch) and semicolon for non-mixers.

Yerms hide two levels of the complexity and yerm-eras hide another two levels.

Eclipse Season Lunar (5+fraction months = mean eclipse season)

Complexity 1: 6/7, 7/8 (Hepton & Octon)

Complexity 2: 13/15, 20/23 (Tzolkinex & Tritos)

Complexity 3: 33/38, 53/61 (Saros & Inex)

Lunisolar Abundance:

Complexity 1: 1/6, 1/5

Complexity 2: 6/31, 7/36

Complexity 3: 13/67, 20/103

Complexity 4 : 97/500; 33/170

Months of 1/13 year as in Victor’s 28/293 and 43/450 calendars:

Complexity 1: 1/11, 1/10

Complexity 2: 2/21, 3/31

Complexity 3: 15/157, 13/136

Complexity 4: 43/450; 28/293

Karl

16(04(03

From: Palmen,
Karl (STFC,RAL,ISIS) Sent: 14 October 2016 13:04To:[hidden email]Subject: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Dear Calendar People

On the May 16 this year in reply to the issue of continued fractions I said:

“I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion
goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.”

I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring
circles will touch without overlapping. See

If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar
is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading
to greater jitter and more difficulty working out a conversion algorithm for the calendar.

Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from
other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion.

I found one measure of structural complexity, which involves Ford circles:

The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia
picture.

Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and
are

… 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, …

These cycles have complexity 1.

The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33-year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years
with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions:

8/33 for solar leap day calendars

2/11 & 3/17 for leap week calendars (but not 5/28)

8/15 & 9/17 for months in pure lunar calendars. These are the yerms in my lunar yerm calendar

3/8 & 4/11 for the leap month years in a lunisolar calendar.

You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19-year Metonic cycle with 7 leap years (touches 3/8) and
the Tabular Islamic calendar (touches 4/11). Another example is the 28-year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions:

Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128)

11/62 for leap week calendars (also 5/28, 8/45 & 14/79)

26/49 for months in for pure lunar calendars (3-yerm cycle)

11/30 for 12-month years in pure lunar calendars (but not 29/79)

7/19 for leap month years in a lunisolar calendar

Complexity 4 includes the 334-year cycle, 353-year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293-year leap week cycle used in the Symmetry454 calendar and also the months of my yerm
calendar. Examples previously mentioned have fractions:

52/293 & 69/389 for leap week calendars (also 41/231)

451/850 for months of my yerm calendar or any other one-era cycle

29/79 for 12-month years in a pure lunar calendars

123/334 & 130/353 for the leap month years of a lunisolar calendar.

Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions:

71/400 & 159/896 for leap week calendars

928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multi-era cycle

239/649, 267/725 & 2519/6840 for the leap month years of lunisolar calendars

In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850)
have complexity 4. This is an example of a mixing addition rule (with yerms mixed) that applies to this measure of complexity.

Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either
of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion.

I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3.

cLc|cL|cLc|cLc|cLc|cL|cLc|cLc|cLc|cL|cLc|

Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3.

cLccLcLc|cLccLccLcLc|cLccLccLcLc|

Both steps of mixing can be shown as

cLc|cL|cLc||cLc|cLc|cL|cLc||cLc|cLc|cL|cLc|

Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1.
I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target.

So a year of such a length would be very bad for a calendar cycle whose leap years are spread as smoothly as possible.

I show each cycle as the fractional part of its mean year in bold and the cycle in ‘c’ common years and ‘L’ leap years with the first year chosen to make the
leap years as late as possible (K=0). The denominator of the fraction is equal to the length of the cycle.

The Ford circle of a cycle is the Ford circle of the fractional part of its mean year, which will serve as notation of the example cycles below.

For example, the 293-year leap week cycle 52/293 has its Ford circle
touch 11/62, which in turn touches 3/17, which in turn touches 1/6, which in turn
touches 0/1. There are 4 touchings, therefore 4 steps and also this is the minimum and so 52/293 has complexity 4. Note that the Ford circles of a/b & c/d touch if and only if ad & bc differ by 1.

A cycle has complexity 1 if it has only one leap year or one common year.

A cycle has complexity 2 if its number of years is one different from a proper multiple of the number of leap years or the number of common years.

So a year of such a length would be very bad for a calendar cycle whose leap years are spread as smoothly as possible.

I show each cycle as the fractional part of its mean year in bold and the cycle in ‘c’ common years and ‘L’ leap years with the first year chosen to make the
leap years as late as possible (K=0). The denominator of the fraction is equal to the length of the cycle.

The Ford circle of a cycle is the Ford circle of the fractional part of its mean year, which will serve as notation of the example cycles below.

For example, the 293-year leap week cycle 52/293 has its Ford circle
touch 11/62, which in turn touches 3/17, which in turn touches 1/6, which in turn
touches 0/1. There are 4 touchings, therefore 4 steps and also this is the minimum and so 52/293 has complexity 4. Note that the Ford circles of a/b & c/d touch if and only if ad & bc differ by 1.

A cycle has complexity 1 if it has only one leap year or one common year.

A cycle has complexity 2 if its number of years is one different from a proper multiple of the number of leap years or the number of common years.

Re: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Dear Walter & Calendar People

This measure of complexity only applies when the leap years are spread as smoothly as possible and so it does not apply to the 128-year cycle as described by Walter.

However if the 128-year cycle were made of four 33-year cycles with one Olympiad removed, the 31 leap years would be spread as smoothly as possible and the complexity is 3 as shown below. The same applies to the 400-year cycle made
of twelve 33-year cycles and one Olympiad.

Complexity 3: 31/128 &71/400

Note that 27 33-year cycles with nine years of two leap years added to make a 900-year cycle of 218 leap years does not have its leap years spread as smoothly as possible.

So a year of such a length would be very bad for a calendar cycle whose leap years are spread as smoothly as possible.

I show each cycle as the fractional part of its mean year in bold and the cycle in ‘c’ common years and ‘L’ leap years with the first year chosen to make the leap years as late as possible (K=0). The denominator of the fraction is
equal to the length of the cycle.

The Ford circle of a cycle is the Ford circle of the fractional part of its mean year, which will serve as notation of the example cycles below.

For example, the 293-year leap week cycle 52/293 has its Ford circle touch 11/62, which in turn touches 3/17, which in turn touches 1/6, which in turn touches 0/1. There are 4 touchings, therefore
4 steps and also this is the minimum and so 52/293 has complexity 4. Note that the Ford circles of a/b & c/d touch if and only if ad & bc differ by 1.

A cycle has complexity 1 if it has only one leap year or one common year.

A cycle has complexity 2 if its number of years is one different from a proper multiple of the number of leap years or the number of common years.

>Note that 27 33-year cycles with nine years >of two leap years added to make a 900->year cycle of 218 leap years does not have >its leap years spread as smoothly as >possible.

It is my impression that "No useful purpose is likely to be served by my inputs 'except opening - pandora of thoughts to defer options - with respect to Reform of the Gregorian calendar LIKE the one I tried to

find ways for implementing in continuation of the existing scheme of things to be of least 'costly to Tax-payer' as the Easiest, Surest and Cheapest ever proposal capable to

block most 'anomalies' in the existing scheme of things. My targets still remain to

demonstrate for Mean Year=(365+31/128)= 365.2421875 days 'possibly closest' to the current Astronomers' Average Mean Year value; and Mean Lunation=29.53058886 i.e. 29d12h44m2s.877504 days on 'slight' increasing the duration of ONE Tithi using 2*(448-years/5541
moons) during 299th-year & 597th-year; of my 896-years/327257 days/11082 moons.

This "adhika/extra time duration" becomes self-consumed as ONE extra moon over about a cycle of Precession, as discussed in my posts, an improvement over Hebrew Lunar 'molad' Calendar [Mean moon= 29d12h44+1/18m].

My tryst with Reforming the Gregorian calendar 'seems to me not finding any headway' - thus leaving for decision for men of Science & Society to "allow or deny" my demonstrated results. Linking of my developed Lunar Tithi value (1 338/326919 day) with
Harappan Era and the 19-year cycle/6932.5 tithi meet current needs for astronomy calculations, demonstrated. The slightly shorter 'tithi of 1 335/326919 day' is an exact fit for 19-years!